The purpose of this material is to derive the equation for the
time rate of change of vorticity in such a way as to point up the role
that the intersection of the pressure levels and density levels plays
in the development of vorticity. For an inertial frame of reference
the equations of motion for a
parcel of air are, in vector form,:

dv/dt = -(1/ρ)∇p -
gk + f

where v is the velocity vector, ρ the density, p pressure, g the
acceleration due to gravity, k the unit
vertical vector and f the
vector of friction forces.
The pressure gradient term

-(1/ρ)∇p

is especially important.

This term can be put into an interesting form by noting that from the
definition of potential temperature θ:

ln(θ) = ln(T) + κln(p0) - κln(p)

and when the gradient operator ∇
is applied to this equation the result is

∇θ/θ = ∇T/T - κ∇p/p

From the defintion of entropy s it follows that

∇s = cp∇θ/θ
and hence from the previous equation

∇s = cp∇T/T - R∇p/p
since κ = R/cp.

Multiplying through by T and noting that cp∇T
is the same as ∇h, where h stands for enthalpy, results in

T∇s - ∇h = -(RT/p)∇p = -(1/ρ)∇p

Thus if the pressure gradient term in the equations of motion is replaced with T∇s - ∇h
the result is

dv/dt = ∂v/∂t + v·∇v

Thus the equations of motion for the atmosphere can be
expressed in vector form as

(1) ∂v/∂t =
v×q +
T∇s − ∇(v2/2 + h + gz) + f

The curl operator ∇× can be
applied to
this equation. The curl of any gradient of a scalar field vanishes;
i.e.,
∇×∇γ=0 for any
scalar field γ because of the equality of cross derivatives.
Therefore under the curl operation ∇(v2/2 + h + gz) vanishes.

Also, because the curl of a curl vanishes,

∇×(T∇×s) = ∇T×∇s.

The result of applying the curl operator to the left-hand side of the above
equation of motion (1) and taking into account the interchangeability of the time and
space derivatives is

∇×(∂v/∂t)
= ∂(∇×v)/∂t
= ∂q/∂t

Equating this to the result of applying the curl operation to the right-hand side of the
equation (1) gives

∂q/∂t
= −∇×(v×q)
+ ∇T×∇s
+ ∇×f

This form of the vorticity equation points out the role of the
intersection or non-intersection of the isothermal surface and the
isoentropic surface through the term
∇T×∇s,
which has a magnitude equal to |∇T||∇s|sin(φ)
where φ is the angle between the two vectors.

Note that since ∇s = cp∇T/T - R∇p/p

∇T×∇s =
∇T×(cp∇T/T - R∇p/p)
= -∇T×(R∇p/p) =
-(R/p)∇T×∇p)

Thus the ∇T×∇s term in the vorticity
equation can be replaced by a term involving
∇p×∇ρ. Generally all
of these cross product terms, called solenoid terms, are proportional and they all vanish
when the atmosphere is barotropic; i.e.; when ∇ρ
always has the same direction as ∇p.